How do you mirror and invert at the same time?
Unfortunately, the mirror modifier can't so this as it creates copies of the object for each axis you tick instead of just transforming it but this is easily done with the Interactive Mirror tool (Ctrl + M) or from the 3d view header, Object > Mirror > Interactive Mirror.

Simply duplicate the half of the mesh you want to mirror and use CtrlM + Xto mirror it on the X axis, then without deselecting, mirror it again but this time on the Y axis.



Only disadvantage as compared to the modifier is that if you update one side, you will have to perform the mirror operations again (this can be avoided if you link both object's Mesh Data with CtrlD > Object Data but adds unnecessary complexity IMO). Also, as per Understanding mirroring and inverting an object, you will have to recalculate the normals if you perform this operation in Edit mode.